arXiv subjects

Joanna A. Ellis-Monaghan

Publications and source records attributed to Joanna A. Ellis-Monaghan.

At least 19 recordsLinked to original sources

arXiv2026

Duality and minors for embeddings of graphs in pseudosurfaces

Cellular embeddings of graphs in surfaces have well-defined duality and minor (edge contraction and deletion) operations that interact in a natural way. A pseudosurface is obtained from a surface (compact 2-manifold) by a finite number of identifications of finite sets of points. Points that are created by the identifications do not have a neighborhood homeomorphic to an open disk and are known as pinchpoints. Embeddings of graphs in pseudosurfaces have been considered, both implicitly and explicitly, since the 1960s. Usually the condition that all pinchpoints correspond to vertices of the graph is imposed. However, this makes it difficult to define duality and minors for pseudosurface embeddings and have these operations interact in the expected way. We define the class of pseudocellular embeddings of graphs in pseudosurfaces, which allow pinchpoints at places other than vertices, in particular in the middle of faces or edges. A subclass known as quasicellular embeddings corresponds to previous embedding models due to Deneen, Shute, and Thomborson and to Huggett and Moffatt. Pseudocellular embeddings also generalize other structures, including the edge-point ribbon graphs of Ellis-Monaghan, Kauffman, and Moffatt, and cyclically ordered graphs or cogs (also known as rigid-vertex graphs). Duality and minor operations for pseudocellular embeddings have very simple and straightforward definitions using topological quotient operations. Pseudocellular embeddings of edgeless graphs have nontrivial structure, and we define some minor operations for those that are related to `t-minor' operations on bipartite graphs. We develop a family of polynomial invariants for pseudocellular embeddings and discuss connections to other polynomial invariants.

math.CO↗

arXiv2026

A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial

We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.

math.CO↗

arXiv2026

A Hypergraph Tutte Polynomial

We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion--contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.

math.CO↗

arXiv2025

Polynomial invariants of cyclically ordered graphs

Cyclically ordered graphs, or cogs, sit between abstract graphs and cellularly embedded graphs. They arise naturally in topological graph theory, knot theory, and mathematical biology. We develop a formal theory of cogs and establish a number of invariants of cogs. In particular we detail several ways to present cogs and detail how these descriptions can be used to construct cog invariants by adapting the matching, transition and Yamada polynomials.

math.CO↗

arXiv2024

Thistlethwaite Theorems for Knotoids and Linkoids

The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollobás-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.

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arXiv2024

Maximum genus embeddings of dense eulerian graphs with specified faces

We give a density condition for when, subject to a necessary parity condition, an eulerian graph or digraph may be cellularly embedded in an orientable surface so that it has exactly two faces, each bounded by an euler circuit, one of which may be specified in advance. More generally, suppose that every vertex in an $n$-vertex eulerian digraph (loops and multiple arcs allowed) has at least $(4n+2)/5$ neighbors, and specify any decomposition of the arcs into disjoint directed circuits (closed trails). We show that such a digraph has an orientable embedding in which the given circuits are facial walks and there are exactly one or two other faces. This embedding then has maximum genus relative to the given circuits being facial walks. When there is only one other face, it is necessarily bounded by an euler circuit. Consequently, if the numbers of vertices and edges have the same parity, a sufficiently dense digraph $D$ with a given directed euler circuit $T$ has an orientable embedding with exactly two faces, each bounded by an euler circuit, one of which is $T$. These results for digraphs give analogous results for graphs as immediate corollaries. The main theorem encompasses several special cases in the literature, such as when the digraph is a tournament.

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arXiv2024

A coarse Tutte polynomial for hypermaps

We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction reductions where the terminal forms consist of isolated vertices. Our Tutte polynomial extends the classical Tutte polynomial of a graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon graph polynomial), and it is a specialization of the transition polynomial via a medial map transformation. We give hypermap duality and partial duality identities for our polynomial, as well as some evaluations, and examine relations between our polynomial and other hypermap polynomials.

math.CO↗

arXiv2024

The Mock Alexander Polynomial for Knotoids and Linkoids

The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for generalizations of knotoids. We prove a conjecture on the mock Alexander polynomial for knotoids, which generalizes to uni-linkoids. Afterwards we give constructions for canonical invariants of linkoids derived from the mock Alexander polynomial, using the formalism of generalized knotoids due to Adams et al.

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arXiv2024

Bi-eulerian embeddings of graphs and digraphs

In 1965 Edmonds showed that every eulerian graph has a bi-eulerian embedding, i.e., an embedding with exactly two faces, each bounded by an euler circuit. We refine this result by giving conditions for a graph to have a bi-eulerian embedding that is specifically orientable or nonorientable. We give connections to the maximum genus problem for directed embeddings of digraphs, in which every face is bounded by a directed circuit. Given an eulerian digraph $D$ with all vertices of degree 2 mod 4 and a directed euler circuit $T$ of $D$, we show that $D$ has an orientable bi-eulerian directed embedding with one of the faces bounded by $T$; this is a maximum genus directed embedding. This result also holds when $D$ has exactly two vertices of degree $0$ mod $4$, provided they are interlaced by $T$. More generally, if $D$ has $\ell$ vertices of degree 0 mod 4, we can find an orientable directed embedding with a face bounded by $T$ and with at most $\ell+1$ other faces. We show that given an eulerian graph $G$ and a circuit decomposition $C$ of $G$, there is an nonorientable embedding of $G$ with the elements of $C$ bounding faces and with one additional face bounded by an euler circuit, unless every block of $G$ is a cycle and $C$ is the collection of cycles of $G$. In particular, every eulerian graph that is not edgeless or a cycle has a nonorientable bi-eulerian embedding with a given euler circuit $T$ bounding one of the faces. Polynomial-time algorithms giving the specified embeddings are implicit in our proofs.

math.CO↗

arXiv2022

Irreducibility of the Tutte polynomial of an embedded graph

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.

math.CO↗

arXiv2022

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings Under Symmetries

Motivated by a problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside's Lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

math.CO↗

arXiv2019

Edge-outer graph embedding and the complexity of the DNA reporter strand problem

In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a DNA reporter strand, that is, a closed walk covering every edge either once or twice, in opposite directions if twice, and passing through each vertex in a particular way. This corresponds to showing that every graph has an \emph{edge-outer embedding}, that is, an orientable embedding with some face that is incident with every edge. In the motivating application, the objective is such a closed walk of minimum length. Here we give a short algorithmic proof of the original existence result, and also prove that finding a shortest length solution is NP-hard, even for $3$-connected cubic ($3$-regular) planar graphs. Independent of the motivating application, this problem opens a new direction in the study of graph embeddings, and we suggest new problems emerging from it.

math.CO↗

arXiv2018

Edge colourings and topological graph polynomials

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.

math.CO↗

arXiv2014

The Las Vergnas Polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas' results on Eulerian circuits from graphs in surfaces of low genus to surfaces of arbitrary genus.

math.CO↗

arXiv2014

A note on recognizing an old friend in a new place: list coloring and the zero-temperature Potts model

Here we observe that list coloring in graph theory coincides with the zero-temperature antiferromagnetic Potts model with an external field. We give a list coloring polynomial that equals the partition function in this case. This is analogous to the well-known connection between the chromatic polynomial and the zero-temperature, zero-field, antiferromagnetic Potts model. The subsequent cross fertilization yields immediate results for the Potts model and suggests new research directions in list coloring.

math.CO↗

arXiv2014

Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.

math.CO↗

arXiv2014

DNA origami and the complexity of Eulerian circuits with turning costs

Building a structure using self-assembly of DNA molecules by origami folding requires finding a route for the scaffolding strand through the desired structure. When the target structure is a 1-complex (or the geometric realization of a graph), an optimal route corresponds to an Eulerian circuit through the graph with minimum turning cost. By showing that it leads to a solution to the 3-SAT problem, we prove that the general problem of finding an optimal route for a scaffolding strand for such structures is NP-hard. We then show that the problem may readily be transformed into a Traveling Salesman Problem (TSP), so that machinery that has been developed for the TSP may be applied to find optimal routes for the scaffolding strand in a DNA origami self-assembly process. We give results for a few special cases, showing for example that the problem remains intractable for graphs with maximum degree 8, but is polynomial time for 4-regular plane graphs if the circuit is restricted to following faces. We conclude with some implications of these results for related problems, such as biomolecular computing and mill routing problems.

math.CO↗

arXiv2012

A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.

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